Math Problem Statement
Solve the following logarithmic equation.
log Subscript 3 Baseline left parenthesis x plus 11 right parenthesis plus log Subscript 3 Baseline left parenthesis x plus 3 right parenthesis equals 2log3(x+11)+log3(x+3)=2
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The solution set is
StartSet nothing EndSetenter your response here.
(Simplify your answer. Type an exact answer. Use a comma to separate answers as needed.
B.
There is no solution.
Solution
We are given the logarithmic equation:
Step 1: Apply the logarithmic property
We can use the property of logarithms that states:
Using this property:
Step 2: Convert the logarithmic equation to an exponential equation
The equation now becomes:
We can rewrite this in exponential form. Since implies , we get:
Step 3: Expand the left-hand side
Expand the left-hand side of the equation:
So the equation becomes:
Step 4: Simplify the equation
Subtract 9 from both sides:
Step 5: Solve the quadratic equation
We solve the quadratic equation:
We can use the quadratic formula:
For the equation , the coefficients are , , and . Plugging these values into the quadratic formula:
Now, calculate the two possible values for :
Step 6: Check for extraneous solutions
Since the logarithm is undefined for negative arguments, we must check whether these values of make the expressions inside the logarithms positive.
- For , and , which are both positive, so this solution is valid.
- For , , which is negative, so this solution is not valid.
Final Answer
The only valid solution is .
The solution set is:
The correct choice is A.
Would you like more details on any part of the process?
Here are 5 questions related to this topic:
- How do you solve logarithmic equations that involve products?
- Can all logarithmic equations be solved by converting to exponential form?
- How do you determine whether a logarithmic solution is extraneous?
- What happens if the argument of a logarithmic function is negative?
- How does the quadratic formula help in solving logarithmic equations?
Tip: Always check for extraneous solutions when solving logarithmic equations, as the logarithm of a negative number is undefined.
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Math Problem Analysis
Mathematical Concepts
Logarithms
Algebra
Quadratic Equations
Exponential Equations
Formulas
log_b(a) + log_b(c) = log_b(a * c)
log_b(a) = c implies b^c = a
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Theorems
Logarithmic properties
Quadratic formula
Suitable Grade Level
Grades 9-12